Could not contracts before inversion if at all: He could not → He couldn't → Couldn't he but He could not → Could he not, and not *Could not he, which is generally ungrammatical, though it's possible for this form to appear in formal language, albeit quite rarely, due to Heavy NP Shift.
– snailboat♦Sep 27 '15 at 2:46

1

And could not have he is impossible. In questions the subject inverts only with the first auxiliary, not with any subsequent complement of that auxiliary.
– StoneyBOct 7 '15 at 21:36

The versions with contractions are definitely less formal. If you're trying to be formal, I would avoid them. Yes, they're a mouthful -- but then again, you're not speaking them, so they're only a penful or a keyboardful.